The clue in this game is just looking at the first two positions of the left apple (and sometimes the last position too!)
For example: If the left apple is S (Super something something) and the middle is H then the right apple has to be A.
If S goes to C(Center), then R(right) and ends in L(Left) the new position of the three apples should be 'SAH'
So the explanation of the writings below is just:
The current position of the apples-> The two first positions of the left apple (the last position of left apple)-> The new positions of the three apples.

SHA->CR(L)->SAH
SHA->CR(C)->HSA
SHA->CR(R)->HAS
SHA->RC->AHS
SHA->CL->ASH
SHA->RL->SHA

AHS->CR(L)->ASH
AHS->CR(C)->HAS
AHS->CR(R)->HSA
AHS->RC->SHA
AHS->CL->SAH
AHS->RL->AHS

ASH->CR(L)->AHS
ASH->CR(C)->SAH
ASH->CR(R)->SHA
ASH->RC->HSA
ASH->CL->HAS
ASH->RL->ASH

HAS->CR(L)->HSA
HAS->CR(C)->AHS
HAS->CR(R)->ASH
HAS->RC->SAH
HAS->CL->SHA
HAS->RL->HAS

HSA->CR(L)->HAS
HSA->CR(C)->SHA
HSA->CR(R)->SAH
HSA->RC->ASH
HSA->CL->AHS
HSA->RL->HSA

SAH->CR(L)->SHA
SAH->CR(C)->ASH
SAH->CR(R)->AHS
SAH->RC->HAS
SAH->CL->HSA
SAH->RL->SAH